with respect to \(m^2\) , it is easy to see that the last statement is valid for the same cases, when the standard prescription \(m^2\rightarrow m^2-i0\)  defines the Feynman propagator. It follows from the quasiclassical asymptotic of the operator \(H(\omega )\) , where all the derivatives entering it are replaced by the momenta \(-ip_\mu \) , that \(\varepsilon ^{\prime }_k(\omega )\)  can be negative for \(\omega >0\)  under the ergosphere, i.e., the antiparticle branch appears in the positive-frequency region. This results in the particle creation [1]} and the reconstruction of the vacuum state. Therefore we restrict our consideration by the region out of the ergosphere.
